hyperbolic circumcircles

 Our aim is to find a condition for the existence of a hyperbolic circle through the vertices of a hyperbolic triangle in terms of the lengths of its sides. We begin with a theorem which holds in both euclidean and hyperbolic geometry, i.e. a theorem of neutral geometry. In the theorem, we consider the size of angles, and assume that each angle is taken in the range (0,π). That is to say, we take 1, G is quadratic in T. Now observe that, if ΔABC has a hyperbolic circumcircle K(P,r), then we can apply the lemma to the hyperbolic centre P. We then have R = S = T = cosh(r), and the condition becomes (R2-1){(R2-Z)2+ (R2-Y)2+ (R2-X)2}-2(R2-Z)(R2-X)(R2-Y)- (R2-1)3 = 0, where X =cosh(a), Y = cosh(b), Z = cosh(c), R = cosh(r). The left hand side is clearly a polynomial in R2, and appears to involve terms in R4 and R6. If we multiply out and simplify, we get the equation F(cosh(a),cosh(b),cosh(c))R2 = Δ2(cosh(a),cosh(b),cosh(c)), where F(X,Y,Z) = 3+2(XY+YZ+ZX)-2(X+Y+Z)-(X2+Y2+Z2), and Δ2(X,Y,Z) = 1+2XYZ-(X2+Y2+Z2). This is the the circumradius equation By expanding then simplifying we get the difference formula Δ2(X,Y,Z)-F(X,Y,Z) = 2(X-1)Y-1)(Z-1). We have met Δ2 before, in connection with the trigonometry of hyperbolic triangles. From that context, we know that it must be positive. We take Δ as the positive root of this quantity. It follows that, as R2 > 0, we must have F(cosh(a),cosh(b),cosh(c)) > 0. We have arrived at the following necessary condition for a hyperbolic circumcircle If the hyperbolic triangle ΔABC has a hyperbolic circumcircle, then (1) F(cosh(a),cosh(b),cosh(c)) > 0, and (2) the hyperbolic radius r is determined by cosh(r) = Δ(cosh(a),cosh(b),cosh(c))/F½(cosh(a),cosh(b),cosh(c)). The other hyperbolic functions of r are easily determined from cosh(r). The manipulation is a little tedious, but we obtain sinh2(r) = 2(cosh(a)-1)cosh(b)-1)(cosh(c)-1)/F(cosh(a),cosh(b),cosh(c)), tanh2(r) = 2(cosh(a)-1)cosh(b)-1)(cosh(c)-1)/Δ2(cosh(a),cosh(b),cosh(c)). The formula for sinh(r) shows again that we must have F > 0. The formula for tanh(r) gives an alternative formulation of the condition Alternative necessary condition If ΔABC has a hyperbolic circumcircle, then we must have 2(cosh(a)-1)cosh(b)-1)(cosh(c)-1) < Δ2(cosh(a),cosh(b),cosh(c)). This is immediate as tanh(r) < 1. If we observe that cosh(x)-1 = 2sinh2(½x), the formula for tanh(r) yields tanh(r) = 4sinh(½a)sinh(½b)sinh(½c)/Δ(cosh(a),cosh(b),cosh(c)) The removal of the need to take a root is an illusion - Δ involves a root. There are various ways to rewrite the formulae for r. Some are given as a digression. Our main purpose is to show that either of our conditions are also sufficient to guarantee the existence of a circumcircle. This is not so easy. We give a proof in a separate page.